SO(3) Special Othorgonal Group

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SUMMARY

The discussion centers on the special orthogonal group SO(3) and its relationship with SO(2). It establishes that the stabilizer of a non-zero vector x in R^3 consists of all rotations about the axis defined by x and -x. This stabilizer is isomorphic to SO(2), which represents all rotations in the orthogonal plane to x. The geometric interpretation confirms that SO(2) encompasses all rotations by any angle in R^2, highlighting the correspondence between the rotations in R^3 and those in the plane orthogonal to the chosen axis.

PREREQUISITES
  • Understanding of special orthogonal groups, specifically SO(3) and SO(2)
  • Familiarity with G-sets and G-actions in group theory
  • Basic knowledge of vector spaces, particularly R^3 and R^2
  • Concept of isomorphism in mathematical structures
NEXT STEPS
  • Study the properties and applications of SO(3) in 3D rotations
  • Explore the geometric interpretations of group actions in G-sets
  • Learn about the relationship between SO(2) and rotations in R^2
  • Investigate isomorphisms in group theory and their implications
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Mathematicians, physicists, and students studying group theory, particularly those interested in rotational symmetries and their applications in three-dimensional spaces.

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For the special orthogonal group SO(3), with G-set R^3, and the usual G-action, we choose x in R^3 not equal to 0. Then the stabilizer of x (set of all the transformations in SO(3) that doesn't change x) is all the rotations about the axis produced by x (and -x). Can someone explain why the stabilizer of x is isomorphic to SO(2)? The notes I have just writes this as if it should be obvious. Am I missing something? Also, just to make sure I'm understanding this right, is the geometric interpretation for SO(2) the group of all rotations by any angle in R^2?
 
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The set of rotations around an axis x in R^3 is isomorphic to the set of rotations in the plane orthogonal to x. And, yes, SO(2) is the set of rotations of a plane.
 

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